Intermediate Microeconomics

Transcription

1 Intermediate Microeconomics Recitation #1 Sai Ma New York University September 10, 2013 Sai Ma (NYU) Recitation #1 September 10, / 21

2 Introduction Sai Ma Second Year Econ PhD o ce hour Depatment of Economics 19 West 4th, 5-8th Floor O ce 819 Sai Ma (NYU) Recitation #1 September 10, / 21

3 Course Tips Check message from blackboard frequently. (Not ed to everyone somtimes) Feel free to me regarding short question. Long question is welcomed but appointment is preferred. Make sure to fully understand the problem sets Help understand the materials Help prepare the midterm and nal Practice and practice Questions from textbook recommanded Sai Ma (NYU) Recitation #1 September 10, / 21

4 Question 1 Problem Joe has the following reference over the co ee shops: for any two co ee shops A,B, he weakly prefers A to B if a small cup of co ee is at least cheap at A as at B, and also A is at least as close (to his house) as B. Thus if p i, d i are the price, distance for co ee shop i, A B () p A p B and d A d B Show that his preference relation may not be complete Sai Ma (NYU) Recitation #1 September 10, / 21

5 Question 1 Let s quickly refresh our memory about the basics of Preference. We use the following symbols to denote an individual s preferences between any two alternatives: % means weak peference means strick peference means indiference Preferences are rational if % satisies two properties: complete: for any two choices A, B, either A % B or B % A or both when doing a questionnaire choosing A, B, the answer "I don t know" is not allowed transitive: if A % B and B % C, then A % C Sai Ma (NYU) Recitation #1 September 10, / 21

6 Question 1 Solution For % to be complete, it must be that for any pair A, B, either A % B or B % A or both.this does not hold if there are two co ee shops A, B such that p A < p B and d A > d B. Why? Sai Ma (NYU) Recitation #1 September 10, / 21

7 Question 1 Solution For % to be complete, it must be that for any pair A, B, either A % B or B % A or both.this does not hold if there are two co ee shops A, B such that p A < p B and d A > d B. Why? The rst inequality implies that B % A does not hold, since B is more expensive; the second implies that A % B does not hold, since A is further away. Sai Ma (NYU) Recitation #1 September 10, / 21

8 Question 2 Problem Explain graphically why indi erence curves cannot cross Sai Ma (NYU) Recitation #1 September 10, / 21

9 Question 2 Let s formally de ne the indi erence curve De nition Let % be a preference relation on a set X.The indi erence curve IC (x) is a set of all y 2 X for which y x In the set notation, IC (x) = fy 2 X jy xg Sai Ma (NYU) Recitation #1 September 10, / 21

10 Question 2 Solution Graph on the Board. Suppose by contradiction that the indi erence curve can cross. Then by IC 2, A B. But according to IC 1, A C and C B. By transitivity, this implies A B, contradiction. Sai Ma (NYU) Recitation #1 September 10, / 21

11 Question 3 Problem (3a) Argue that the utility unction v(x 1, x 2 ) = 2 ln x 1 + ln x 2 represents the same preference as u(x 1, x 2 ) = x x Problem (3b) Write the equation of an indi erence curve which represents these preferences and goes through the point (1,1) Problem (3c) If his consumer has twice as many units of good 1 as good 2, then what s his marginal rate of substitution (MRS) between the goods? Sai Ma (NYU) Recitation #1 September 10, / 21

12 Question 3 Let s brie y talk about utility function. De nition We say that the function U : X! R represents the preference % if for all x and y 2 X, x % y if and only i U(x) > U(y).If the function U represents the preference relation %, we refer to it has a utility function or we say that % has a utility representation. Sai Ma (NYU) Recitation #1 September 10, / 21

13 Question 3 Theorem If U represents %, then fo any strictly increasing function f : R! R, the function V (x) = f (U(x)) represents % as well Proof? Sai Ma (NYU) Recitation #1 September 10, / 21

14 Question 3 Proof. x % y () U(x) > U(y) (since U represents % ) () f (U(x)) > f (U(y)) (since f is strictly increasing) () V (x) > U(y) Sai Ma (NYU) Recitation #1 September 10, / 21

15 Question 3 Back to our problem Problem (3a) Argue that the utility unction v(x 1, x 2 ) = 2 ln x 1 + ln x 2 represents the same preference as u(x 1, x 2 ) = x x Solution Here v = ln(u 3 ).This is a strictly increasing function of u (why?)and hence by the theorem, v represents the same preference. Sai Ma (NYU) Recitation #1 September 10, / 21

16 Question 3 Problem (3b) Write the equation of an indi erence curve which represents these preferences and goes through the point (1,1) Solution We want all points (x 1, x 2 ) satisfying (x 1, x 2 ) (1, 1) : since u represents the preferences, this requires u(x 1, x 2 ) = u(1, 1), x x = 1 Note you can also use v, in that case the equation is 2 ln x 1 + ln x 2 = 0 (verify) Graph? Sai Ma (NYU) Recitation #1 September 10, / 21

17 Question 3 Problem (3b) Write the equation of an indi erence curve which represents these preferences and goes through the point (1,1) Solution We want all points (x 1, x 2 ) satisfying (x 1, x 2 ) (1, 1) : since u represents the preferences, this requires u(x 1, x 2 ) = u(1, 1), x x = 1 Note you can also use v, in that case the equation is 2 ln x 1 + ln x 2 = 0 (verify) Graph? plot x 2 = 1 x 2 1 Sai Ma (NYU) Recitation #1 September 10, / 21

18 Question 3 For 3(c), let s review marginal rate of substitution (MRS) tells you the rate at which you would give up y to get a bit more x Formula in this case: MRS (x,y ) = MU x MU y Negative slope of the indi erence curve Diminishing MRS implies preferences are convex Sai Ma (NYU) Recitation #1 September 10, / 21

19 Question 3 Problem (3c) If his consumer has twice as many units of good 1 as good 2, then what s his marginal rate of substitution (MRS) between the goods? Solution MU 1 = u x 1 = 2 3 (x 2 x 1 ) 1 3 MU 2 = u x 2 = 1 3 (x 1 x 2 ) 2 3 So, MRS (x1,x 2 ) = MU 1 MU 2 = 2x 2 x 1 ) MRS (x1,x 2 ) = 1 when x 1 = 2x 2 Mush less tedious if using v(x 1, x 2 ) instead of u. Sai Ma (NYU) Recitation #1 September 10, / 21

20 Question 4 Problem John always eats his ballpark hotdog in a special way: he uses a footlong hotdog with precisely half a bun, 1 ounce of mustard, and 2 ounces of pickle relish. His utility depends only on these four items, and any extra amount of a single item (without the others) is worthless to him. What form does his utility function have? Solution? Sai Ma (NYU) Recitation #1 September 10, / 21

21 Question 4 Problem John always eats his ballpark hotdog in a special way: he uses a footlong hotdog with precisely half a bun, 1 ounce of mustard, and 2 ounces of pickle relish. His utility depends only on these four items, and any extra amount of a single item (without the others) is worthless to him. What form does his utility function have? Solution? u(h, b, m, r) = min(h, 2b, m, 0.5r) Sai Ma (NYU) Recitation #1 September 10, / 21

22 Question 4 Problem John always eats his ballpark hotdog in a special way: he uses a footlong hotdog with precisely half a bun, 1 ounce of mustard, and 2 ounces of pickle relish. His utility depends only on these four items, and any extra amount of a single item (without the others) is worthless to him. What form does his utility function have? Solution? u(h, b, m, r) = min(h, 2b, m, 0.5r) (1, 0.5, 1, 2) (1, 0.5, 800, 2) Sai Ma (NYU) Recitation #1 September 10, / 21

23 Convexity De nitions Preferences % are convex if x % y implies αx + (1 α)y % y, 8α 2 (0, 1). De nitions Preferences % are strictly convex if a % y, b % y and a 6= bimplies αa + (1 α)b y, 8α 2 (0, 1). Sai Ma (NYU) Recitation #1 September 10, / 21

24 Convexity In the footnote in chapter 3 of textbook, there is a fancy way to check if the preference is convex. Denote f be the utility function, then the represented preference is strictly convex if f 2 2 f 11 2f 1 f 2 f 12 + f 2 1 f 22 < 0 So if the cross derivative is zero (i.e f 12 = 0), then the preference is strictly convex if f 11 < 0 and f 22 < 0 Examples (Cobb-Douglas) u(x, y) = ln x + ln y (Perfect Substitution) u(x, y) = x + y (CES) u(x, y) = x δ +y δ δ You can always use diminishing MRS criterion to check if the preferences are convex Sai Ma (NYU) Recitation #1 September 10, / 21

25 CES Utility Function u(x, y) = x δ +y δ δ If δ = 1, u(x, y) = x + y, (perfect substitution) If δ = 0, u(x, y) = ln x + ln y, (Cobb-Douglas). Derivation using L Hôpital s rule If δ =, u(x, y) = min(x, y) perfect complement Derivation beyond the scope of this course Sai Ma (NYU) Recitation #1 September 10, / 21